The proof relies on the following explicit version of Bourgain's Theorem C from [1]}, communicated to us by Misha Rudnev. Let \(E_+(A, \xi A)\)  be the number of solutions to \(a_1+\xi a_2 = a_3 + \xi a_4\)  with the \(a_i \in A\)  and \(E_+(A) = E_+(A, A)\) . For each \(\xi \ne 0\) , we have the inequality \(E_+(A, \xi A) \le |A|^3\) . Rudnev's theorem gives strong quantitative bounds on the statement that the \(|A|^3\)  order of magnitude is attained for only few \(\xi \) .
